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Theorem foprcl 4000
Description: Closure law for an operation.
Hypothesis
Ref Expression
foprcl.1 |- F:(R X. S)-->C
Assertion
Ref Expression
foprcl |- ((A e. R /\ B e. S) -> (AFB) e. C)
Proof of Theorem foprcl
StepHypRef Expression
1 foprcl.1 . . 3 |- F:(R X. S)-->C
2 ffnoprval 3999 . . . 4 |- (F:(R X. S)-->C <-> (F Fn (R X. S) /\ A.x e. R A.y e. S (xFy) e. C))
32pm3.27bi 326 . . 3 |- (F:(R X. S)-->C -> A.x e. R A.y e. S (xFy) e. C)
41, 3ax-mp 7 . 2 |- A.x e. R A.y e. S (xFy) e. C
5 opreq1 3953 . . . 4 |- (x = A -> (xFy) = (AFy))
65eleq1d 1532 . . 3 |- (x = A -> ((xFy) e. C <-> (AFy) e. C))
7 opreq2 3954 . . . 4 |- (y = B -> (AFy) = (AFB))
87eleq1d 1532 . . 3 |- (y = B -> ((AFy) e. C <-> (AFB) e. C))
96, 8rcla42v 1871 . 2 |- ((A e. R /\ B e. S) -> (A.x e. R A.y e. S (xFy) e. C -> (AFB) e. C))
104, 9mpi 44 1 |- ((A e. R /\ B e. S) -> (AFB) e. C)
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637   X. cxp 3158   Fn wfn 3167  -->wf 3168  (class class class)co 3948
This theorem is referenced by:  axaddcl 5243  axmulcl 5245  issubgi 8059  ablmul 8068  hvaddclt 8803  hvmulclt 8804  hiclt 8868  iooirrsa 10379
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950
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